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Machine Learning Summer School on Theory and Practice of Computational Learning

Geometric Methods and Manifold Learning

author: Mikhail Belkin, Department of Computer Science and Engineering, Ohio State University
author: Partha Niyogi, Department of Computer Science, University of Chicago
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Slides
0:00 Geometric Methods and Manifold Learning
2:37 High Dimensional Data
9:00 Geometry and Data: The Central Dogma
11:08 Manifold Learning
17:15 Formal Justification
22:28 Take Home Message
23:17 Principal Components Analysis
27:45 Manifold Model
28:52 An Acoustic Example (1)
30:44 An Acoustic Example (2)
31:52 Solutions
32:56 Acoustic Phonetics
34:41 Vision Example
38:00 Robotics
39:38 Manifold Learning
40:08 Differential Geometry
40:10 Manifold Learning
40:49 Differential Geometry
41:02 Embedded manifolds
41:46 Tangent space
42:24 Tangent vectors and curves (1)
42:41 Tangent vectors and curves (2)
43:41 Tangent vectors as derivatives (1)
43:57 Tangent vectors as derivatives (2)
45:30 Riemannian geometry
46:21 Length of curves and geodesics
47:25 Gradient
48:52 Exponential map
50:27 Laplace-Beltrami operator
56:21 Intrinsic Curvature
57:10 Dimensionality Reduction
64:37 Algorithmic framework (1)
65:24 Algorithmic framework (2)
66:04 Algorithmic framework (3)
66:20 Isomap
68:34 Multidimensional Scaling (1)
70:56 Multidimensional Scaling (2)
72:04 Isomap
73:32 Unfolding flat manifolds
74:36 Locally Linear Embedding (1)
75:48 Locally Linear Embedding (2)
76:54 Laplacian and LLE
77:41 Laplacian Eigenmaps (1)
78:41 Laplacian Eigenmaps (2)
81:24 Laplacian Eigenmaps (3)
82:38 Diffusion Distance
85:28 Diffusion Maps
87:29 Justification
90:09 A Fundamental Identity
90:44 Embedding
90:51 PCA versus Laplacian Eigenmaps
93:18 On the Manifold
94:25 Curves on Manifolds
94:26 Stokes Theorem
94:58 On the Manifold
95:23 Curves on Manifolds
95:26 Stokes Theorem
98:43 Manifold Laplacian
99:29 Properties of Laplacian
100:44 The Circle: An Example
105:07 From graphs to manifolds (1)
105:10 From graphs to manifolds (2)
105:12 Estimating Dimension from Laplacian

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